Lecture 6

Fundamental Algorithms (2 of 2)

(includes text and images from the 3rd edition of the VTK book)

Tensor Algorithms

I'm not even going to pretend that I can talk with any authority on the topic of tensor algorithms.

here are some web pages with info about tensors:

http://mathworld.wolfram.com/Tensor.html
http://bartok.ucsc.edu/peter/114A/tensors.pdf

visualizing 3x3 real symmetric tensors

• describe state of displacement or stress in 3D material
• stress tensor - stress is measure of loading - force per unit area (N / m^2)
• strain tensor - strain is measure of displacement - change in length over length (dL / L)
• diagonal coefficients are normal stresses and strains
  • act perpendicular to a surface
  • normal stress is either compression or tension
• off-diagonal coefficients are shear stresses and strains
  • act tangentially to a surface
• 3x3 real symmetric matrix can be characterized by eigenvectors and eigenvalues
  • eigenvectors - 3 multually perpendicular vectors in 3D - principal axis
  • eigenvalues - 3 numbers
  • bunch of math to get eigenvalues and eigenvectors (http://www.sosmath.com/matrix/eigen0/eigen0.html or http://mathworld.wolfram.com/Eigenvalue.html)
• ordering the eigenvalues largest to smallest allows us to name the eigenvectors
  • major, medium, and minor in the same ordering
• visualization through tensor ellipsoids
  • position a sphere at tensor location
  • eigenvectors form axis (major, medium, minor) of an ellipsoid
  • eigenvalues used to scale the sphere/elipsoid
  • http://www.iua.upf.es/activitats/semirec/semi-rderiche/semi-rderiche/semi-r7.gif
• visualization through tensor axes 
• instead of ellipsoid, just show the principal axes

create or change dataset topology or geometry

Source Objects

• Modelling simple geometry
  • generate sphere, cone, cube
  • read in model files with more complex geometry
• Supporting Geometry
  • additional geometry to support/enhance understanding of the visualization - here is code to create the office airflow visualization including very simple furniture http://web.cs.wpi.edu/~matt/courses/cs563/talks/vtk/office.html
    
• Data attribute creation
  • procedures to create data and attributes
  • implicit functions F(x, y, z) = constant
    • properties:
      • simple geometric description
      • region separation (inside [ F<0], on [F=0], outside [F>0])
      • scalar generation to convert point in space into a scalar value
    • uses:
      • modelling objects
        • can sample F on a dataset and generate an isosurface at contour value c_i
        • can perform boolean operations (union, intersection, difference ) eg vtk ice cream cone
          
        • evaluate F on a regular array/volume of points then generate a scalar value at each point
      • selecting data
        • can select or extract data by choosing cells, points within a particular region. Cell is within a region if all its points are within the region. Evaluate each point and check its sign
        • can perform boolean operations
      • visualizing mathematical descriptions
        • for functions that are not implicit may be able to generate scalar values
  • implicit modelling
    • similar to modelling above except scalars generated using distance function rather than implicit function
    • distance function computed as Euclidean distance to set of generating primitives (point, line, polygon)
    • isosurface values are distances to generating primitive
    • see figure 6-26
    • can perform boolean operations
    • eg hello.tcl
      
    • some nice slides (with bad choices of color) on this here: http://www.cpsc.ucalgary.ca/~jungle/555/notes/implicit.pdf
  • glyphs
    • object/icon that is affected by input data
    • may be geometry, image, dataset
    • may orient, scale, translate, deform etc in response to input data
  • cutting
    • cut through a dataset with a surface & display interpolated data values on the surface
    • can view data on nearly arbitrary surfaces
    • cutting surface defined by an implicit function
    • evaluate F(x, y, z) for each point of each cell in the dataset
    • if not all points + or - then surface passes through the cell
    • generate isosurface F(x, y, z) = 0
    • interpolate cut edges to get data attribute values
    • can make multiple cuts with different isovalues then show them with transparency back to front to do volume rendering

Volume Rendering (Chap 7 vtk)

Have data such as MRI or CT scans, ultrasound, etc

Direct Rendering (no intermediate geometric representation) vs Geometric Rendering

Image order vs object order

• image order - rays cast through each pixel in image plane through volume
• object order - volume traversed back-to-front (usually) processing each voxel
• hybrids

Image Order - raycasting / raytracing

• greyscale data values
  • minimum value = transparent black
  • maximum value = white
• determine values encountered along ray
• processing those values according to ray function - many possibilities
• volume is 3D image dataset w/ scalar values at points of a regular grid
• need an interpolation function to sample locations between the grid points
  • zero-order / constant / nearest neighbour returns closest grid point - simple
  • trilinear - linear interpolation between closest points along 3 axis
• can sample volume at uniform intervals while traversing the ray
  • need to take care in choosing step size - speed vs accuracy tradeoff
• or examine each voxel encountered while traversing the ray (fig 7-8)
  • convert the ray into a discrete form to obtain an ordered sequence of voxels
  • each voxel has 6 sides, 8 vertices, 12 edges
    • 6-connected if each share a face (best at finding small details)
    • 18-connected if share a face or an edge
    • 26-connected if share a face or an edge or a vertex (faster) (fig 7-10)
• ray: (x, y, z) = (xo, yo, zo) + (a, b, c) t where (a, b, c) is normalized direction vector

4 different ray functions - UL:maximum, UR:average, BL:distance=30, BR:composite

a nice program to explore this kind of data on the mac is osirix - http://www.osirix-viewer.com/

Object Order

• typically voxels traversed back to front or front to back and composited using alpha values
• similar to situation where everything in scene is a transparent polygon
• some algorithms do not need to process voxels in a particular order
• typically have triple nested loop:
  • for each plane ordered back to front
    • for each row of the plane
      • for each voxel in the row
        • determine its projected location on the view plane
        • alter the pixel at that location

• problem may occur in that by projecting back to the view plane we may have 'holes' as adjacent voxels may not map onto adjacent pixels
• solution is Splatting - distribute energy of voxel across multiple pixels in a splat/footprint
  • kernel with finite extent placed around each data sample
  • footprint is projection onto the view plane
  • in parallel projection with symmetric kernel the footprint is identical for all samples aside from its location so fast
  • type of kernel, radius of kernel, resolution of footprint table all affect results

• texture mapping volume rendering - based on hardware in the graphics cards/boards
  • 2D or 3D
  • project a set of texture mapped polygons which span the entire volume

  1. sampling step using interpolation (multiple methods, hardware/software)
  2. blending step to combine samples in the frame buffer

  

  Volume rendering using 2D (left) and 3D (right) techniques

  • 2D
    • dataset decomposed into orthographic slices along axis of volume most parallel to the viewing direction
    • move through slices back to front downloading 2D texture into memory, projected
    • may need to interpolate between slices
    • performance based on
      • software sampling rate
      • texture download rate
      • texture scan conversion rate
  • 3D
    • if enough texture memory, entire volume download into texture memory
    • series of planes parallel to view plane along viewing direction generated
    • polygons generated and projected back to front (fig 7-14)
    • 3D interpolation is done on the card/board
    • if not enough texture memory, break the dataset into subvolumes (bricks)
    • where each brick can fit in texture memory
    • process bricks back to front being careful of the boundaries

2D Texture-mapped volume rendering with different opacity mappings

Volume Classification

• transfer function maps information at voxel location into material / colour / opacity
  • eg differentiating air, muscle, bone in a CT scan
• can have separate functions for red, green, blue, opacity

Regions of Interest

• often need to look through parts of the volume to see other parts of the volume
• can specify a partial region of the volume to render that contains interesting things
  • use clipping planes or other geometric forms to specify the region

Coming Next Time

Information Visualization